I'd like to turn this sum: \begin{align}\sum_{n=0}^{\infty} \frac{x^{n+1}}{3^{n+1}(n+1)} \end{align} into an integral $\displaystyle \int_{a}^{b} g(x) \space dx$.
There seems to be many methods to either change or approximate sums as integrals. So I've become confused which approach would work.
In Is it possible to write a sum as an integral to solve it? robjohn used $\int_0^\infty e^{-nt}\,\mathrm{d}t=\frac1n$ which looks similar to a Laplace Transforms.
I can't see how he gets rid of the n's so I'm not able to apply it here otherwise it seems promising. But looking elsewhere there are also approximations methods such as: Turning infinite sum into integral which even more obscure at least to me.
How do I convert this sum to an integral?